1 Overview of Results

Hypothesis testing is arguably one of the most fundamental primitives in quantum information theory. As such it has found many applications, e.g., in quantum channel coding [27] and quantum illumination [37, 46, 56], or for giving an operational interpretation to abstract quantities [13, 16, 28]. A particular hypothesis testing setting is that of quantum state discrimination where quantum states are assigned to each of the hypotheses and we aim to determine which state is actually given. Several distinct scenarios are of interest, which differ in the priority given to different types of error or in how many copies of a system are given to aid the discrimination. Here, we investigate the setting of asymmetric hypothesis testing where the goal is to discriminate between two n-party quantum states (strategies or hypotheses) \(\rho _n\) and \(\sigma _n\) living on the n-fold tensor product of some finite-dimensional inner product space \({\mathcal {H}}^{\otimes n}\). That is, we are optimizing over all two-outcome positive operator valued measures (POVMs) with \(\{M_n,(1-M_n)\}\) and associate \(M_n\) with accepting \(\rho _n\) as well as \(\left( 1-M_n\right) \) with accepting \(\sigma _n\). This naturally gives rise to the two possible errors

$$\begin{aligned}&\alpha _n(M_n):={{\,\mathrm{Tr}\,}}\big [\rho _n(1-M_n)\big ]\;&\text {Type 1 error,} \end{aligned}$$
(1)
$$\begin{aligned}&\beta _n(M_n):= {{\,\mathrm{Tr}\,}}\big [\sigma _nM_n\big ]\;&\text {Type 2 error.} \end{aligned}$$
(2)

For asymmetric hypothesis testing we minimize the Type 2 error asFootnote 1

$$\begin{aligned} \beta (n,\varepsilon ):=\inf _{0\ll M_n\ll 1}\Big \{\beta _n(M_n)\big |\alpha _n(M_n)\le \varepsilon \Big \} \end{aligned}$$
(3)

while we require the Type 1 error not to exceed a small constant \(\varepsilon \in (0,1)\). We are then interested in finding the optimal error exponentFootnote 2

$$\begin{aligned} \zeta (n,\varepsilon ):=-\frac{\log \beta (n,\varepsilon )}{n}, \end{aligned}$$
(4)

and its asymptotic limits

$$\begin{aligned} \zeta (\infty ,\varepsilon ):=\lim _{n\rightarrow \infty }-\frac{\log \beta (n,\varepsilon )}{n},\quad \zeta (\infty ,0):=\lim _{\varepsilon \rightarrow 0}\zeta (\infty ,\varepsilon )\,. \end{aligned}$$
(5)

A well studied discrimination setting is that between fixed independent and identical (iid) states \(\rho ^{\otimes n}\) and \(\sigma ^{\otimes n}\), where the asymptotic error exponent is determined by the quantum Stein’s lemma [4, 30, 43] in terms of the quantum relative entropy. Namely, we denote this special case of Eq. (5) by \(\zeta _{\rho ,\sigma }(\infty ,\varepsilon )\) and the Stein’s lemma then gives for any \(\varepsilon \in (0,1)\) the formula

$$\begin{aligned} \zeta _{\rho ,\sigma }(\infty ,\varepsilon )= D(\rho \Vert \sigma ):={\left\{ \begin{array}{ll} {{\,\mathrm{Tr}\,}}\big [\rho \left( \log \rho -\log \sigma \right) \big ] \quad &{} {{\,\mathrm{supp}\,}}(\rho )\subseteq {{\,\mathrm{supp}\,}}(\sigma )\\ \infty &{} \text {otherwise.}\end{array}\right. } \end{aligned}$$
(6)

In many applications we aim to solve more general discrimination problems and a prominent example of such is that of composite hypotheses—in which we attempt to discriminate between different sets of states. Previously the case of composite iid null hypotheses \(\rho ^{\otimes n}\) with \(\rho \in {\mathcal {S}}\) and corresponding asymptotic error exponent \(\zeta _{{\mathcal {S}},\sigma }(\infty ,\varepsilon )\) was studied in [10, 25], leading to the formula

$$\begin{aligned} \zeta _{{\mathcal {S}},\sigma }(\infty ,\varepsilon )=\inf _{\rho \in S} D(\rho \Vert \sigma )\quad \forall \varepsilon \in (0,1)\,. \end{aligned}$$
(7)

On the other hand, the problem of composite alternative hypotheses is more involved in the non-commutative case. When the set of alternative hypotheses \({\mathcal {T}}_n\) for \(n\in {\mathbb {N}}\) fulfils certain axioms motivated by the framework of resource theories, it was shown in [13] that the corresponding asymptotic error exponent \(\zeta _{\rho ,{\mathcal {T}}}(\infty ,\varepsilon )\) is written in terms of the regularized relative entropy distance as

$$\begin{aligned} \zeta _{\rho ,{\mathcal {T}}}(\infty ,\varepsilon )=\lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\sigma _n\in {\mathcal {T}}_n} D\left( \rho ^{\otimes n}\Vert \sigma _n\right) \quad \forall \varepsilon \in (0,1)\,. \end{aligned}$$
(8)

This regularization is in general needed as we know from the case of the relative entropy of entanglement [54]. Note that this might not be too surprising since the set of alternative hypotheses is not required to be iid in general.

For our main result, we consider the setting where null and alternative hypotheses are both composite and given by convex combinations of n-fold tensor powers of states from given convex, closed sets \({\mathcal {S}}\) and \({\mathcal {T}}\). More precisely, for \(n\in {\mathbb {N}}\) we attempt the following discrimination problem.Footnote 3

Null hypothesis::

the convex hull of iid states

$$\begin{aligned} {\mathcal {S}}_n:=\Big \{\int \rho ^{\otimes n}\;\mathrm {d}\nu (\rho )\Big |\nu \in {\mathcal {S}}\Big \} \text { with }{\mathcal {S}}\subseteq S({\mathcal {H}}) \text { convex and closed} \end{aligned}$$
(9)
Alternative hypothesis::

the convex hull of iid states

$$\begin{aligned} {\mathcal {T}}_n:=\Big \{\int \sigma ^{\otimes n}\;\mathrm {d}\mu (\sigma )\Big |\mu \in {\mathcal {T}}\Big \} \text { with }{\mathcal {T}}\subseteq S({\mathcal {H}}) \text { convex and closed} \end{aligned}$$
(10)

Slightly abusing the notation, \(\nu \in {\mathcal {S}}\) and \(\mu \in {\mathcal {T}}\) stand for probability measures on the Borel \(\sigma \)-algebra of \({\mathcal {S}}\) and \({\mathcal {T}}\), respectively. For \(\varepsilon \in (0,1)\) the goal is to determine the optimal error exponent for composite asymmetric hypothesis testing

$$\begin{aligned} \zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon ):=-\frac{1}{n}\log \inf _{0\ll M_n\ll 1}\left\{ \sup _{\mu \in {\mathcal {T}}}{{\,\mathrm{Tr}\,}}\big [M_n\sigma _n(\mu )\big ]\bigg |\sup _{\nu \in {\mathcal {S}}}{{\,\mathrm{Tr}\,}}\big [(1-M_n)\rho _n(\nu )\big ]\le \varepsilon \right\} \end{aligned}$$
(11)

with the abbreviations

$$\begin{aligned} \rho _n(\nu ):=\int \rho ^{\otimes n}\mathrm {d}\nu (\rho )\quad \text {and}\quad \sigma _n(\mu ):=\int \sigma ^{\otimes n}\mathrm {d}\mu (\sigma )\,. \end{aligned}$$
(12)

It is trivial to see that \(\zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )\) equivalently gives the error exponent of testing between \({\mathcal {S}}^{\otimes n} := \{ \rho ^{\otimes n} | \rho \in {\mathcal {S}}\}\) and \({\mathcal {T}}^{\otimes n} := \{ \sigma ^{\otimes n} | \sigma \in {\mathcal {T}}\}\). This then explicitly takes the form of an iid problem. The following is our main result, which we prove in Sect. 2 under the support condition

$$\begin{aligned} \mathrm {supp}(\rho ) \subseteq \mathrm {supp}(\sigma )\quad \forall \rho \in {\mathcal {S}}\quad \forall \sigma \in {\mathcal {T}}\,. \end{aligned}$$
(13)

Theorem 1.1

For the discrimination problem as above, we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\liminf _{n\rightarrow \infty }\zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )&=\lim _{\varepsilon \rightarrow 0}\limsup _{n\rightarrow \infty }\zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon ) \end{aligned}$$
(14)
$$\begin{aligned}&=\lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}D\Big (\rho ^{\otimes n}\Big \Vert \int \sigma ^{\otimes n}\;\mathrm {d}\mu (\sigma )\Big ) \end{aligned}$$
(15)

Our proof can be found in Sect. 2 and has a clear structure in the sense that we start from the composite Stein’s lemma for classical probability distributions and then lift the result to the non-commutative setting by using elementary properties of entropic measures. We emphasise that even in the case of a fixed null hypothesis \({\mathcal {S}}=\{\rho \}\) our setting is not a special case of the previous results [13], as our sets of alternative hypotheses are not closed under tensor product

$$\begin{aligned} \sigma _m\in {\mathcal {T}}_m,\;\sigma _n\in {\mathcal {T}}_n\nRightarrow \sigma _m\otimes \sigma _n\in {\mathcal {T}}_{mn}\,, \end{aligned}$$
(16)

which is one of the properties required for the results in [13].

We show that in contrast to the finite classical case [11, 34], the regularization in Eq. (15) is needed in general. That is, we provide an explicit example for which the non-regularized relative entropy formula is not an achievable asymptotic error exponent

$$\begin{aligned} \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \sigma \in {\mathcal {T}} \end{array}}D(\rho \Vert \sigma )>\lim _{\varepsilon \rightarrow 0}\limsup _{n\rightarrow \infty }\zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )\,. \end{aligned}$$
(17)

In particular, we find that, even for \(n\rightarrow \infty \), in general

$$\begin{aligned} \frac{1}{n}\inf _{\mu \in {\mathcal {T}}}D\Big (\rho ^{\otimes n}\Big \Vert \int \sigma ^{\otimes n}\;\mathrm {d}\mu (\sigma )\Big )\ne \inf _{\sigma \in {\mathcal {T}}}D(\rho \Vert \sigma )\,, \end{aligned}$$
(18)

thereby providing a counterexample to this conjectured quantum entropy inequality (see [12, Equation (20)] for a variant) which holds in the finite classical setting (see, e.g., [51, Lemma 3.11]).Footnote 4 Note that the \(\le \) direction in Eq. (18) holds trivially.

Nevertheless, there exist non-commutative cases in which the regularization in Eq. (15) is not needed and we discuss several such examples. In particular, we give an operational interpretation of the relative entropy of coherence in terms of hypothesis testing.

Finally, we apply the techniques developed in this work to strengthen previously known quantum relative entropy lower bounds on the conditional quantum mutual information [9, 12, 22, 32, 50, 51, 55]

$$\begin{aligned} I(A:B|C)_\rho :=H(AC)_\rho +H(BC)_\rho -H(ABC)_\rho -H(C)_\rho \end{aligned}$$
(19)

with \(H(C)_\rho :=-{{\,\mathrm{Tr}\,}}\left[ \rho _C\log \rho _C\right] \) the von Neumann entropy. We find that

$$\begin{aligned} I(A:B|C)_\rho \ge \limsup _{n\rightarrow \infty }\frac{1}{n}D\Big (\rho _{ABC}^{\otimes n}\Big \Vert \int \beta _0(t)\;\mathrm {d}t\big ({\mathcal {I}}_A\otimes {\mathcal {R}}^{[t]}_{C\rightarrow BC}(\rho _{AC})\big )^{\otimes n}\Big ) \end{aligned}$$
(20)

for some universal probability distribution \(\beta _0(t)\) and the rotated Petz recovery maps \(R^{[t]}_{C\rightarrow BC}\) as defined in Sect. 4. In contrast to the previously known bounds in terms of the quantum relative entropy [12, 51], the recovery map in Eq. (20) takes a specific form only depending on the reduced state on BC. Note that the regularization in Eq. (20) cannot go away in relative entropy distance, as recently shown in [21]. We end with an overview how all known recoverability lower bounds on the conditional quantum mutual information compare and argue that Eq. (20) represents the last possible strengthening.

The remainder of the paper is structured as follows. In Sect. 2 we prove our main result about composite asymmetric hypothesis testing. This is followed by Sect. 3 where we discuss several concrete examples including an operational interpretation of the relative entropy of coherence, as well as its Rényi analogues in terms of the Petz divergences [44] and the sandwiched relative entropies [39, 57]. In Sect. 4 we prove the refined lower bound on the conditional mutual information from Eq. (20) and use it to show that the regularization in Eq. (15) is needed in general. Finally, we end in Sect. 5 with a discussion of some open questions.

2 Proof of Main Result

In the following we give a proof of our main result Theorem 1.1. We first prove the converse, meaning the \(\le \) direction of Theorem 1.1, which follows from the following proposition.

Proposition 2.1

For \(\rho \in {\mathcal {S}}\), \(\mu \in {\mathcal {T}}\), and \(\varepsilon \in (0,1)\) we have

$$\begin{aligned} \zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon ) \le \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}} \frac{1}{n}\frac{D\left( \rho ^{\otimes n}\Vert \sigma _n(\mu )\right) +1}{1-\varepsilon }\,. \end{aligned}$$
(21)

Proof

We follow the original converse proof of quantum Stein’s lemma [30] for the states \(\rho ^{\otimes n}\) and \(\sigma _n(\mu )\). By the monotonicity of the quantum relative entropy [36], we have for the measurement \(\{M_n,(1-M_n)\}\) that

$$\begin{aligned}&D\left( \rho ^{\otimes n}\big \Vert \sigma _n(\mu )\right) \nonumber \\&\quad \ge {{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \log \frac{{{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] }{{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n(\mu )\right] }+\left( 1-{{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \right) \log \frac{1-{{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] }{1-{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n(\mu )\right] } \end{aligned}$$
(22)
$$\begin{aligned}&\quad \ge -\log 2-{{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \log {{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n(\mu )\right] \end{aligned}$$
(23)
$$\begin{aligned}&\quad \ge -1-\inf _{\rho \in {\mathcal {S}}}{{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \log \sup _{\mu \in {\mathcal {T}}}{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n(\mu )\right] \end{aligned}$$
(24)
$$\begin{aligned}&\quad \ge -1-(1-\varepsilon )\log \sup _{\mu \in {\mathcal {T}}}{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n(\mu )\right] \end{aligned}$$
(25)

leading to

$$\begin{aligned} -\frac{1}{n}\log \sup _{\mu \in {\mathcal {T}}}{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n(\mu )\right] \le \frac{1}{n} \frac{D\left( \rho ^{\otimes n}\big \Vert \sigma _n(\mu )\right) +1}{1-\varepsilon } \end{aligned}$$
(26)

for any \(\rho \in {\mathcal {S}}\), \(\mu \in {\mathcal {T}}\), and \(0\ll M_n \ll 1\) such that \(\sup _{\rho \in {\mathcal {S}}}{{\,\mathrm{Tr}\,}}\left[ (1-M_n)\rho ^{\otimes n}\right] \le \varepsilon \). Taking the supremum over all such \(M_n\) and then the infimum over \(\rho \in {\mathcal {S}}\) and \(\mu \in {\mathcal {T}}\) leads to the desired result. \(\square \)

By taking the appropriate limits in Proposition 2.1, we immediately find the converse statements

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \limsup _{n\rightarrow \infty } \zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )&\le \limsup _{n\rightarrow \infty } \frac{1}{n} \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}} D\left( \rho ^{\otimes n}\big \Vert \sigma _n(\mu )\right) \end{aligned}$$
(27)
$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \liminf _{n\rightarrow \infty } \zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )&\le \liminf _{n\rightarrow \infty } \frac{1}{n} \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}} D\left( \rho ^{\otimes n}\big \Vert \sigma _n(\mu )\right) \, . \end{aligned}$$
(28)

Remark 2.2

In information-theoretic language Eq. (21) represents a weak converse, i.e. the limit \(\varepsilon \rightarrow 0\) in Eqs. (27) and (28) is required, and one might be tempted to derive a strong converse that holds for all \(\varepsilon \in (0,1)\) by employing quantum versions of the Rényi relative entropies [39, 44, 57] or the smooth max-relative entropy [18, 31]. However, because of the missing \(\sigma _m\in {\mathcal {T}}_m,\;\sigma _n\in {\mathcal {T}}_n\nRightarrow \sigma _m\otimes \sigma _n\in {\mathcal {T}}_{mn}\) property, the convergence of aforementioned measures to the quantum relative entropy remains unclear (see, e.g., [3, 13, 17, 19] for corresponding techniques in the context of quantum hypothesis testing). As such, we leave open the question about a strong converse.

For the proof of the achievability, meaning the \(\ge \) direction in Theorem 1.1, the basic idea is to start from the corresponding composite Stein’s lemma for classical probability distributions and lift the result to the non-commutative setting by solely using properties of quantum entropy. For that we need the measured relative entropy defined as [20, 30]

$$\begin{aligned} D_{{\mathcal {M}}}(\rho \Vert \sigma ):=\sup _{({\mathcal {X}},{\mathcal {M}})}D\Big (\underbrace{\sum _{x\in {\mathcal {X}}}{{\,\mathrm{Tr}\,}}\left[ M_x\rho \right] |x\rangle \langle x|}_{=\,{\mathcal {M}}(\rho )}\Big \Vert \underbrace{\sum _{x\in {\mathcal {X}}}{{\,\mathrm{Tr}\,}}\left[ M_x\sigma \right] |x\rangle \langle x|}_{=\,{\mathcal {M}}(\sigma )}\Big )\,, \end{aligned}$$
(29)

where the optimization is over finite sets \({\mathcal {X}}\) and measurements \({\mathcal {M}}\) on \({\mathcal {X}}\) with \({{\,\mathrm{Tr}\,}}\left[ M_x\rho \right] \) a measure on \({\mathcal {X}}\). Henceforth, we write for the classical relative entropy between probability distributions \(D(P\Vert Q)\)—defined via the diagonal embedding of P and Q as on the right-hand side of Eq. (29). It is known that we can restrict the a priori unbounded supremum to rank-one projective measurements [7, Theorem 2]. We now prove the achievability direction in Theorem 1.1 in several steps and start with an achievability bound in terms of the measured relative entropy.

Lemma 2.3

For definitions as above and \(\varepsilon \in (0,1)\), we have

$$\begin{aligned} \liminf _{n\rightarrow \infty }\zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )\ge \sup _{k\in {\mathbb {N}}}\frac{1}{k}\inf _{\begin{array}{c} \nu \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}D_{{\mathcal {M}}}\left( \rho _k(\nu )\Vert \sigma _k(\mu )\right) \,. \end{aligned}$$
(30)

Proof

For sets of classical probability distributions \({\mathcal {S}}\) and \({\mathcal {T}}\), we get from the corresponding commutative achievability result that for \(\delta >0\) and \(\varepsilon \in (0,1)\), there exists \(M_{\varepsilon ,\delta }\in {\mathbb {N}}\) such that for \(m\ge M_{\varepsilon ,\delta }\) we have

$$\begin{aligned} \zeta _{{\mathcal {S}},{\mathcal {T}}}(m,\varepsilon )\ge \inf _{\begin{array}{c} P\in {\mathcal {S}}\\ Q\in {\mathcal {T}} \end{array}}D(P\Vert Q)-\delta \,. \end{aligned}$$
(31)

This is a special case of [11, Theorem 2] and we refer to [34] as well as references therein for a general discussion of composite hypothesis testing. Now, the strategy is to first measure the quantum states and then to invoke the classical achievability result from Eq. (31) for the resulting probability distributions.

This argument is made precise as follows. The classical case implies the existence of a sequence of tests \((T_{k, m})_{m\in {\mathbb {N}}}\) for the discrimination problem between the measured state \({\mathcal {M}}_k({\mathcal {S}}^{\otimes k})^{\otimes m}\) and the measured state \({\mathcal {M}}_k({\mathcal {T}}^{\otimes k})^{\otimes m}\) with \(m\in {\mathbb {N}}\), such that

$$\begin{aligned} \sup _{\rho \in {\mathcal {S}}} {{\,\mathrm{Tr}\,}}\left[ (1 - T_{k, m}) {\mathcal {M}}_k( \rho ^{\otimes k})^{\otimes m}\right] \le \varepsilon \end{aligned}$$
(32)

for all \(m\in {\mathbb {N}}\), and

$$\begin{aligned} \lim _{m\rightarrow \infty } - \frac{1}{m} \log \sup _{\sigma \in {\mathcal {T}}} {{\,\mathrm{Tr}\,}}\left[ T_{k, m} {\mathcal {M}}_k(\sigma ^{\otimes k})^{\otimes m}\right] \ge \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \sigma \in {\mathcal {T}} \end{array}} D\left( {\mathcal {M}}_k(\rho ^{\otimes k})\big \Vert {\mathcal {M}}_k(\sigma ^{\otimes k}\right) \,. \end{aligned}$$
(33)

Hence, for any \(\delta >0\), there exists an \(m_\delta \) such that for all \(m\ge m_\delta \) we have

$$\begin{aligned} - \frac{1}{m} \log \sup _{\sigma \in {\mathcal {T}}} {{\,\mathrm{Tr}\,}}\left[ T_{k, m} {\mathcal {M}}_k(\sigma ^{\otimes k})^{\otimes m}\right] \ge \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \sigma \in {\mathcal {T}} \end{array}} D\left( {\mathcal {M}}_k(\rho ^{\otimes k}) \big \Vert {\mathcal {M}}_k(\sigma ^{\otimes k}\right) - \delta \,. \end{aligned}$$
(34)

Defining \(T_n := \big ({\mathcal {M}}_k^\dagger \big )^{\otimes m}(T_{k,m})\otimes 1_r\) for \(n=km+r\), \(r\in \{0,\dots , k-1\}\), we get that

$$\begin{aligned} \sup _{\rho \in {\mathcal {S}}} {{\,\mathrm{Tr}\,}}\left[ (1- T_n)\rho ^{\otimes n}\right] = \sup _{\rho \in {\mathcal {S}}} {{\,\mathrm{Tr}\,}}\left[ (1 - T_{k,m}) {\mathcal {M}}_k(\rho ^{\otimes k})^{\otimes m}\right] \le \varepsilon \end{aligned}$$
(35)

for all \(n\in {\mathbb {N}}\), and thus

$$\begin{aligned} \zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )&\ge -\frac{1}{n} \log \sup _{\sigma \in {\mathcal {T}}} {{\,\mathrm{Tr}\,}}\left[ T_n\sigma ^{\otimes n}\right] \end{aligned}$$
(36)
$$\begin{aligned}&= -\frac{1}{km+r}\log \sup _{\sigma \in {\mathcal {T}}} {{\,\mathrm{Tr}\,}}\left[ T_{k,m} {\mathcal {M}}_k(\sigma ^{\otimes k})^{\otimes m}\right] \end{aligned}$$
(37)
$$\begin{aligned}&\ge \frac{m}{km+r} \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \sigma \in {\mathcal {T}} \end{array}} D\left( {\mathcal {M}}_k(\rho ^{\otimes k})\big \Vert {\mathcal {M}}_k(\sigma ^{\otimes k})\right) - \frac{m}{km+r} \delta \end{aligned}$$
(38)

whenever \(n\ge km_\delta \). Therefore, we get

$$\begin{aligned} \liminf _{n\rightarrow \infty } \zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon ) \ge \frac{1}{k} \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \sigma \in {\mathcal {T}} \end{array}} D\left( {\mathcal {M}}_k(\rho ^{\otimes k}) \big \Vert {\mathcal {M}}_k(\sigma ^{\otimes k})\right) - \frac{1}{k} \delta \end{aligned}$$
(39)

for any binary POVM \({\mathcal {M}}_k\) and \(\delta >0\). Taking \(\delta \rightarrow 0\) and then the supremum over \({\mathcal {M}}_k\) gives

$$\begin{aligned} \liminf _{n\rightarrow \infty } \zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )&\ge \frac{1}{k} \sup _{{\mathcal {M}}_k} \inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \sigma \in {\mathcal {T}} \end{array}} D({\mathcal {M}}_k(\rho ^{\otimes k}) \Vert {\mathcal {M}}_k(\sigma ^{\otimes k})) \end{aligned}$$
(40)
$$\begin{aligned}&\ge \frac{1}{k} \sup _{{\mathcal {M}}_k} \inf _{\begin{array}{c} \nu \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}} D({\mathcal {M}}_k(\rho _k(\nu )) \Vert {\mathcal {M}}_k(\sigma _k(\mu ))) \end{aligned}$$
(41)
$$\begin{aligned}&= \frac{1}{k} \inf _{\begin{array}{c} \nu \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}} \sup _{{\mathcal {M}}_k} D({\mathcal {M}}_k(\rho _k(\nu )) \Vert {\mathcal {M}}_k(\sigma _k(\mu )))\,, \end{aligned}$$
(42)

where the equality follows from Lemma A.2. Since this holds for every \(k\in {\mathbb {N}}\), we find the claimed

$$\begin{aligned} \liminf _{n\rightarrow \infty } \zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )&\ge \sup _{k\in {\mathbb {N}}} \frac{1}{k} \inf _{\begin{array}{c} \nu \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}} D_{\mathcal {M}}(\rho _k(\nu ) \Vert \sigma _k(\mu ))\,. \end{aligned}$$
(43)

\(\square \)

Next, we argue that the measured relative entropy can in fact be replaced by the quantum relative entropy by only paying an asymptotically vanishing penalty term. For this we need the following lemma, which can be seen as a generalization of the technical argument in the original proof of quantum Stein’s lemma [30].

Lemma 2.4

Let \(\rho _n,\sigma _n\in S\left( {\mathcal {H}}^{\otimes n}\right) \) with \(\sigma _n\) permutation invariant. Then, we have

$$\begin{aligned} D\big (\rho _n\big \Vert \sigma _n\big )-\log {{\,\mathrm{poly}\,}}(n)\le D_{{\mathcal {M}}}\big (\rho _n\big \Vert \sigma _n\big )\le D\big (\rho _n\big \Vert \sigma _n\big )\,, \end{aligned}$$
(44)

where \({{\,\mathrm{poly}\,}}(n)\) stands for terms of order at most polynomial in n.

Proof

We can restrict ourselves to the case where \({{\,\mathrm{supp}\,}}\big (\rho _n\big )\subseteq {{\,\mathrm{supp}\,}}\big (\sigma _n\big )\) since otherwise all relative entropy terms evaluate to infinity by definition. The second inequality follows directly from the definition of the measured relative entropy in Eq. (29) together with the fact that the quantum relative entropy is monotone [36]. We now prove the first inequality with the help of asymptotic spectral pinching [25]. The pinching map with respect to \(\omega \in S({\mathcal {H}})\) is defined as

$$\begin{aligned} {\mathcal {P}}_\omega (\cdot ):=\sum _{\lambda \in \mathrm {spec}(\omega )} P_\lambda (\cdot )P_\lambda \;\text {with the spectral decomposition } \omega =\sum _{\lambda \in \mathrm {spec}(\omega )}\lambda P_\lambda . \end{aligned}$$
(45)

Crucially, we have the pinching operator inequality [25]

$$\begin{aligned} {\mathcal {P}}_\omega [X]\gg \frac{X}{|\mathrm {spec}(\omega )|}\,, \end{aligned}$$
(46)

where \(|\mathrm {spec}(\cdot )|\) denotes the size of the spectrum. From this we can deduce that (see, e.g., [52, Lemma 4.4])

$$\begin{aligned} D\big (\rho _n\big \Vert \sigma _n\big )-\log \big |\mathrm {spec}\big (\sigma _n\big )\big |\le D\big ({\mathcal {P}}_{\sigma _n}\big (\rho _n\big )\big \Vert \sigma _n\big ) \le D_{{\mathcal {M}}}\big (\rho _n\big \Vert \sigma _n\big )\,, \end{aligned}$$
(47)

where the second inequality follows since \({\mathcal {P}}_{\sigma _n}\big (\rho _n\big )\) and \(\sigma _n\) are diagonal in the same basis and the measured relative entropy gives an upper-bound. It remains to show that \(\big |\mathrm {spec}\big (\sigma _n\big )\big |\le {{\,\mathrm{poly}\,}}(n)\). However, since \(\sigma _n\) is permutation invariant we have by Schur-Weyl duality (see, e.g., [24, Section 5]) that in the Schur basis

$$\begin{aligned} \sigma _n=\bigoplus _{\lambda \in \Lambda _n}\sigma _{Q_\lambda }\otimes 1_{P_\lambda }\quad \text {with } |\Lambda _n|\le {{\,\mathrm{poly}\,}}(n) \text { and }\text {dim} \left[ \sigma _{Q_\lambda }^0\right] \le {{\,\mathrm{poly}\,}}(n). \end{aligned}$$
(48)

where \(\sigma _{Q_\lambda }^0\) is the projector onto the support of \(\sigma _{Q_\lambda }\). This implies the claim. \(\square \)

By combining Lemma 2.3 together with Lemma 2.4 we find for \(\varepsilon \in (0,1)\) that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon )\ge \limsup _{n\rightarrow \infty }\frac{1}{n}\inf _{\begin{array}{c} \nu \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}D\left( \rho _n(\nu )\Vert \sigma _n(\mu )\right) \,. \end{aligned}$$
(49)

The next step is to argue that asymptotically the infimum over states \(\rho _n(\nu )\) can without loss of generality be restricted to iid states \(\rho ^{\otimes n}\) with \(\rho \in {\mathcal {S}}\).

Lemma 2.5

For definitions as above and \(\omega _n\in S\left( {\mathcal {H}}^{\otimes n}\right) \), we have

$$\begin{aligned} \frac{1}{n}\inf _{\nu \in {\mathcal {S}}}D\left( \rho _n(\nu )\big \Vert \omega _n\right) \ge \frac{1}{n}\inf _{\rho \in {\mathcal {S}}}D\left( \rho ^{\otimes n}\big \Vert \omega _n\right) -\frac{2d^2\log (n+1)}{n}\,, \end{aligned}$$
(50)

where \(d:=\mathrm {dim}\left( {\mathcal {H}}\right) \).

Proof

For \(\nu \in {\mathcal {S}}\) and \(H(\rho ):=-{{\,\mathrm{Tr}\,}}\left[ \rho \log \rho \right] \) the von Neumann entropy, we observe the following chain of arguments

$$\begin{aligned}&\frac{1}{n} D\left( \rho _n(\nu )\big \Vert \omega _n\right) \nonumber \\&= \frac{1}{n} D\Big (\sum _{i=1}^Np_i\rho _i^{\otimes n}\Big \Vert \omega _n\Big ) \end{aligned}$$
(51)
$$\begin{aligned}&=-\frac{1}{n}H\Big (\sum _{i=1}^Np_i\rho _i^{\otimes n}\Big )-\frac{1}{n}\sum _{i=1}^Np_i{{\,\mathrm{Tr}\,}}\left[ \rho _i^{\otimes n}\log {\omega _n}\right] \end{aligned}$$
(52)
$$\begin{aligned}&\ge -\frac{1}{n}\sum _{i=1}^N p_i H\left( \rho _i^{\otimes n}\right) -\frac{\log {(n+1)^{2d^2}}}{n}-\frac{1}{n}\sum _{i=1}^Np_i{{\,\mathrm{Tr}\,}}\left[ \rho _i^{\otimes n}\log {\omega _n}\right] \end{aligned}$$
(53)
$$\begin{aligned}&\ge \min _{\rho _i}\frac{1}{n}D\left( \rho ^{\otimes n}_i\big \Vert \omega _n\right) -\frac{2d^2\log (n+1)}{n} \end{aligned}$$
(54)
$$\begin{aligned}&\ge \inf _{\rho \in {\mathcal {S}}}\frac{1}{n}D\left( \rho ^{\otimes n}\big \Vert \omega _n\right) -\frac{2d^2\log (n+1)}{n}\,, \end{aligned}$$
(55)

where the first equality holds by an application of Carathédory’s theorem with \(N\le (n+1)^{2d^2}\) (Lemma A.3), and the first inequality by an almost-convexity property of the von Neumann entropy (Lemma A.4). All other steps are elementary. Since the above argument holds for all \(\nu \in {\mathcal {S}}\), the claim follows. \(\square \)

Lemma 2.5 together with Eq. (49) gives for \(\varepsilon \rightarrow 0\) that

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}D\left( \rho ^{\otimes n} \Vert \sigma _n(\mu )\right)&\le \sup _{k\in {\mathbb {N}}} \frac{1}{k} \inf _{\begin{array}{c} \nu \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}} D_{\mathcal {M}}(\rho _k(\nu ) \Vert \sigma _k(\mu )) \end{aligned}$$
(56)
$$\begin{aligned}&\le \lim _{\varepsilon \rightarrow 0} \liminf _{n\rightarrow \infty }\zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon ) \end{aligned}$$
(57)
$$\begin{aligned}&\le \liminf _{n\rightarrow \infty }\frac{1}{n}\inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}D\left( \rho ^{\otimes n} \Vert \sigma _n(\mu )\right) \,, \end{aligned}$$
(58)

where the last step follows from Eq. (28). This shows that the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}D\left( \rho ^{\otimes n} \Vert \sigma _n(\mu )\right) \end{aligned}$$
(59)

exists and all the inequalities above hold as equalities. Furthermore, we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}D\left( \rho ^{\otimes n} \Vert \sigma _n(\mu )\right)&\le \lim _{\varepsilon \rightarrow 0}\limsup _{n\rightarrow \infty }\zeta _{{\mathcal {S}},{\mathcal {T}}}(n,\varepsilon ) \end{aligned}$$
(60)
$$\begin{aligned}&\le \limsup _{n\rightarrow \infty }\frac{1}{n}\inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}D\left( \rho ^{\otimes n} \Vert \sigma _n(\mu )\right) \,, \end{aligned}$$
(61)

which concludes the proof of Theorem 1.1. \(\square \)

3 Examples and Extensions

Here, we discuss several concrete examples of composite discrimination problems—some of which have a single-letter solution.

3.1 Relative entropy of coherence

Following the literature around [5], the set of states diagonal in a fixed basis \(\{|c\rangle \}\) is called incoherent and denoted by \({\mathcal {C}}\subseteq S({\mathcal {H}})\). The relative entropy of coherence of \(\rho \in S({\mathcal {H}})\) is defined as

$$\begin{aligned} D_{{\mathcal {C}}}(\rho ):=\inf _{\sigma \in {\mathcal {C}}} D(\rho \Vert \sigma )\,. \end{aligned}$$
(62)

Based on our main result (Theorem 1.1), we can characterize the following discrimination problem.

Null hypothesis::

the fixed state \(\rho ^{\otimes n}\)

Alternative hypothesis::

the convex hull of iid coherent states

$$\begin{aligned} {\bar{{\mathcal {C}}}}_n:=\Big \{\int \sigma ^{\otimes n}\;\mathrm {d}\mu (\sigma )\Big |\mu \in {\mathcal {C}}\Big \} \end{aligned}$$
(63)

Namely, Theorem 1.1 gives

$$\begin{aligned} \zeta _{{\bar{{\mathcal {C}}}}}(\infty ,0):=&\lim _{\varepsilon \rightarrow 0}\lim _{n\rightarrow \infty }\zeta _{{\bar{{\mathcal {C}}}}}(n,\varepsilon ) \end{aligned}$$
(64)
$$\begin{aligned} =&\lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\mu \in {\mathcal {C}}}D\Big (\rho ^{\otimes n}\Big \Vert \int \sigma ^{\otimes n}\;\mathrm {d}\mu (\sigma )\Big ) \end{aligned}$$
(65)
$$\begin{aligned} =&D_{{\mathcal {C}}}(\rho )\,, \end{aligned}$$
(66)

where the limit in Eq. (64) exists because the relative entropy of coherence is additive on product states [14], and the last step follows from a general property of the relative entropy of coherence (Lemma A.5) applied to the decohering channel. In fact, there is even a single-letter solution for the following less restricted discrimination problem.

Null hypothesis::

the fixed state \(\rho ^{\otimes n}\)

Alternative hypothesis::

the convex set of coherent states \({\mathcal {C}}_n\)

It is straightforward to check that this hypothesis testing problem fits the general framework of [13], leading to

$$\begin{aligned} \zeta _{{\mathcal {C}}}(\infty ,\varepsilon ):=\lim _{n\rightarrow \infty }\zeta _{{\mathcal {C}}}(n,\varepsilon )=\lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\sigma _n\in {\mathcal {C}}_n}D\left( \rho ^{\otimes n}\Vert \sigma _n\right) =D_{{\mathcal {C}}}(\rho )\quad \forall \varepsilon \in (0,1)\,, \end{aligned}$$
(67)

where the last step again follows from a general property of the relative entropy of coherence (Lemma A.5). Thus, we have two a priori different hypothesis testing scenarios that both give an operational interpretation to the relative entropy of coherence. In the following we give a simple self-contained proof of Eq. (67) that is different from the rather involved steps in [13] and instead follows ideas from [4, 28]. The goal is the quantification of the optimal asymptotic error exponent

$$\begin{aligned} \zeta _{{\mathcal {C}}}(n,\varepsilon )&:=-\frac{1}{n}\log \inf _{\begin{array}{c} 0\ll M_n\ll 1\\ {{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \ge 1-\varepsilon \end{array}}\sup _{\sigma _n\in {\mathcal {C}}_n}{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n\right] \end{aligned}$$
(68)
$$\begin{aligned} \mathrm {with}\quad \zeta _{{\mathcal {C}}}(\infty ,\varepsilon )&:=\lim _{n\rightarrow \infty }\zeta _{{\mathcal {C}}}(n,\varepsilon )\,. \end{aligned}$$
(69)

Proposition 3.1

For the discrimination problem as above with \(\varepsilon \in (0,1)\), we have

$$\begin{aligned} \zeta _{{\mathcal {C}}}(\infty ,\varepsilon )=D_{{\mathcal {C}}}(\rho )\,. \end{aligned}$$
(70)

Note that Proposition 3.1 is independent of \(\text {supp}(\rho )\) as the set \({\mathcal {C}}\) includes full rank states. A weak converse for \(\varepsilon \rightarrow 0\) follows exactly as in Lemma 2.1, together with Lemma A.5 to make the expression single-letter. For the strong converse as claimed in Proposition 3.1, we make use of a general family of quantum Rényi entropies: the Petz divergences [44]. For \(\rho ,\sigma \in S({\mathcal {H}})\) and \(s\in (0,1)\cup (1,\infty )\) they are defined as

$$\begin{aligned} D_s\left( \rho \big \Vert \sigma \right) :=\frac{1}{s-1}\log {{\,\mathrm{Tr}\,}}\left[ \rho ^s\sigma ^{1-s}\right] \,, \end{aligned}$$
(71)

whenever either \(s<1\) and \(\rho \) is not orthogonal to \(\sigma \) in Hilbert-Schmidt inner product or \(s>1\) and the support of \(\rho \) is contained in the support of \(\sigma \). (Otherwise we set \(D_s(\rho \Vert \sigma ):=\infty \).) The corresponding Rényi relative entropies of coherence are given by [14]

$$\begin{aligned} D_{s,{\mathcal {C}}}(\rho ):=\inf _{\sigma \in {\mathcal {C}}}D_s(\rho \Vert \sigma ) \text { with the additivity property }D_{s,{\mathcal {C}}}\left( \rho ^{\otimes n}\right) =n D_{s,{\mathcal {C}}}(\rho ). \end{aligned}$$
(72)

Using similar standard arguments [40] as in Lemma 2.1 but based on the monotonicity of the Petz divergences, we find for \(s\in (1,2]\) that

$$\begin{aligned}&-\frac{1}{n}\log \inf _{0\le M_n\le 1}\Big \{{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n\right] \Big |{{\,\mathrm{Tr}\,}}\left[ (1-M_n)\rho ^{\otimes n}\right] \le \varepsilon \Big \}\nonumber \\&\quad \le \frac{1}{n}\cdot D_s\left( \rho ^{\otimes n}\big \Vert \sigma _n\right) +\frac{1}{n}\frac{s}{s-1}\frac{1}{\log (1-\varepsilon )}\,. \end{aligned}$$
(73)

By taking the infimum over \(\sigma _n\in {\mathcal {C}}_n\), a basic application of Sion’s minimax theorem (Lemma A.1), using the additivity from Eq. (72), taking the limit \(n\rightarrow \infty \) as well as the limit [14]

$$\begin{aligned} \lim _{s\rightarrow 1}D_{s,{\mathcal {C}}}(\rho )=D_{{\mathcal {C}}}(\rho )\,, \end{aligned}$$
(74)

we find the claimed strong converse \(\zeta _{{\mathcal {C}}}(\infty ,\varepsilon )\le D_{{\mathcal {C}}}(\rho )\). The achievability direction of Proposition 3.1 is based on the Petz divergences as well.

Lemma 3.2

For the discrimination problems as above with \(n\in {\mathbb {N}}\) and \(\varepsilon \in (0,1)\), we have for \(s\in (0,1)\) that

$$\begin{aligned} \zeta _{{\mathcal {C}}}(n,\varepsilon )\ge D_{s,{\mathcal {C}}}(\rho )-\frac{1}{n}\frac{s}{1-s}\log \frac{1}{\varepsilon }\,. \end{aligned}$$
(75)

Taking the limit \(n\rightarrow \infty \) as well as the limit \(s\rightarrow 1\) using Eq. (74), we then find the claimed achievability \(\zeta _{{\mathcal {C}}}(\infty ,\varepsilon )\ge D_{{\mathcal {C}}}(\rho )\).

Proof of Lemma 3.2

It is straightforward to check with Sion’s minimax theorem (Lemma A.1) that

$$\begin{aligned} \inf _{\begin{array}{c} 0\le M_n\le 1\\ {{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \ge 1-\varepsilon \end{array}}\sup _{\sigma _n\in {\mathcal {C}}_n}{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n\right] =\sup _{\sigma _n\in {\mathcal {C}}_n}\inf _{\begin{array}{c} 0\le M_n\le 1\\ {{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \ge 1-\varepsilon \end{array}}{{\,\mathrm{Tr}\,}}\left[ M_n\sigma _n\right] \,. \end{aligned}$$
(76)

Now, for \(\lambda _n\in {\mathbb {R}}\) with \(n\in {\mathbb {N}}\) we choose \(M_n(\lambda _n):=\left\{ \rho ^{\otimes n}-2^{\lambda _n}\sigma _n\right\} _+\) where \(\{\cdot \}_+\) denotes the projector on the eigenspace of the positive spectrum. We have \(0\ll M_n(\lambda _n)\ll 1\) and by Audenaert’s inequality (Lemma A.6) with \(s\in (0,1)\) we get

$$\begin{aligned} {{\,\mathrm{Tr}\,}}\left[ (1-M_n(\lambda _n))\rho ^{\otimes n}\right] \le 2^{(1-s)\lambda _n}{{\,\mathrm{Tr}\,}}\left[ \left( \rho ^{\otimes n}\right) ^s\sigma _n^{1-s}\right] =2^{(1-s)\left( \lambda _n-D_s\left( \rho ^{\otimes n}\big \Vert \sigma _n\right) \right) }\,. \end{aligned}$$
(77)

Moreover, again Audenaert’s inequality (Lemma A.6) for \(s\in (0,1)\) implies

$$\begin{aligned} {{\,\mathrm{Tr}\,}}\left[ M_n(\lambda _n)\sigma _n\right] \le 2^{-s\lambda _n}{{\,\mathrm{Tr}\,}}\left[ \left( \rho ^{\otimes n}\right) ^s\sigma _n^{1-s}\right] =2^{-s\lambda _n-(1-s)D_s\left( \rho ^{\otimes n}\big \Vert \sigma _n\right) }\,. \end{aligned}$$
(78)

Hence, choosing

$$\begin{aligned} \lambda _n:=D_s\left( \rho ^{\otimes n}\big \Vert \sigma _n\right) +\log \varepsilon ^{\frac{1}{1-s}} \text { with } M_n:=M_n(\lambda _n) \end{aligned}$$
(79)

leads with Eq. (77) to \({{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \ge 1-\varepsilon \). Finally, Eq. (76) together with Eq. (78) and the additivity property from Eq. (72) leads to the claim.

We note that a more refined analysis of the above calculation allows to determine the Hoeffding bound as well as the strong converse exponent (cf. [4, 28]). The former gives an operational interpretation to the Rényi relative entropy of coherence \(D_{s,{\mathcal {C}}}(\rho )\), whereas the latter gives an operational interpretation to the sandwiched Rényi relative entropies of coherence [14]

$$\begin{aligned} {\tilde{D}}_{s,{\mathcal {C}}}(\rho ):=\inf _{\sigma \in {\mathcal {C}}}{\tilde{D}}_s(\rho \Vert \sigma ) \end{aligned}$$
(80)

with the sandwiched Rényi entropies

$$\begin{aligned} {\tilde{D}}_s(\rho \Vert \sigma ):=\frac{1}{s-1}\log {{\,\mathrm{Tr}\,}}\left[ \left( \sigma ^{\frac{1-s}{2s}}\rho \sigma ^{\frac{1-s}{2s}}\right) ^s\right] \end{aligned}$$
(81)

whenever either \(s<1\) and \(\rho \) is not orthogonal to \(\sigma \) in Hilbert-Schmidt inner product or \(s>1\) and the support of \(\rho \) is contained in the support of \(\sigma \) [39, 57]. (Otherwise we set \(D_s(\rho \Vert \sigma ):=\infty \).) The crucial insight for the proof is again the additivity property \({\tilde{D}}_{s,{\mathcal {C}}}\left( \rho ^{\otimes n}\right) =n{\tilde{D}}_{s,{\mathcal {C}}}(\rho )\), that was already shown in [14].

3.2 Relative entropy of recovery

The relative entropy of recovery of \(\rho _{ABC}\in S({\mathcal {H}}_{ABC})\) and its regularized version are defined as [7, 12, 47]Footnote 5

$$\begin{aligned} D(A;B|C)_\rho :=&\inf _{{\mathcal {R}}} D\big (\rho _{ABC}\big \Vert ({\mathcal {I}}_{A}\otimes {\mathcal {R}}_{C\rightarrow BC})\left( \rho _{AC}\right) \big ) \end{aligned}$$
(82)
$$\begin{aligned} \mathrm {and}\quad D^{\infty }(A;B|C)_\rho :=&\lim _{n\rightarrow \infty }\frac{1}{n}D(A;B|C)_{\rho ^{\otimes n}}\,, \end{aligned}$$
(83)

where the infimum goes over all completely positive and trace preserving maps \({\mathcal {R}}_{C\rightarrow BC}\). It was recently shown that in general [21]

$$\begin{aligned} D^{\infty }(A;B|C)_\rho \ne D(A;B|C)_\rho \,. \end{aligned}$$
(84)

Using the framework from [13], the following discrimination problem was linked to the regularized relative entropy of recovery [16].

Null hypothesis::

the fixed state \(\rho _{ABC}^{\otimes n}\)

Alternative hypothesis::

for any \({\mathcal {R}}_{C^n\rightarrow B^nC^n}\) completely positive and trace preserving, the convex set of states

$$\begin{aligned} {\mathcal {R}}^n:=\left\{ ({\mathcal {I}}_{A^n}\otimes {\mathcal {R}}_{C^n\rightarrow B^nC^n})\left( \rho _{AC}^{\otimes n}\right) \right\} \end{aligned}$$
(85)

Namely, for \(\varepsilon \in (0,1)\) we have for the corresponding asymptotic error exponent

$$\begin{aligned} \zeta _{{\mathcal {R}}}(\infty ,\varepsilon ):=\lim _{n\rightarrow \infty }\zeta _{{\mathcal {R}}}(n,\varepsilon )=D^{\infty }(A;B|C)_\rho \,. \end{aligned}$$
(86)

In contrast, our main result (Theorem 1.1) covers the following discrimination problem.

Null hypothesis::

the fixed state \(\rho _{ABC}^{\otimes n}\)

Alternative hypothesis::

for any \({\mathcal {R}}_{C\rightarrow BC}\) completely positive and trace preserving, the convex hull of iid states

$$\begin{aligned} {\bar{{\mathcal {R}}}}^n:=\Big \{\int \left( ({\mathcal {I}}_A\otimes {\mathcal {R}}_{C\rightarrow BC})(\rho _{AC})\right) ^{\otimes n}\;\mathrm {d}\mu ({\mathcal {R}})\Big \}\,. \end{aligned}$$
(87)

Interestingly, we can show that the asymptotic error exponents of the two discrimination problems are actually identical.

Proposition 3.3

With the definitions as above, we have

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n}\inf _{{\mathcal {R}}} D\big (\rho _{ABC}^{\otimes n}\big \Vert ({\mathcal {I}}_A\otimes {\mathcal {R}}_{C^n\rightarrow B^nC^n})\left( \rho _{AC}^{\otimes n}\right) \big ) \nonumber \\&\quad = \lim _{n\rightarrow \infty } \frac{1}{n} \inf _{\mu \in {\mathcal {R}}} D\Big (\rho _{ABC}^{\otimes n}\Big \Vert \int \big (({\mathcal {I}}_A\otimes {\mathcal {R}}_{C\rightarrow BC})(\rho _{AC})\big )^{\otimes n}\;\mathrm {d}\mu ({\mathcal {R}})\Big )\,. \end{aligned}$$
(88)

Proof

One direction of the inequality is by definition and for the other direction we use a de Finetti reduction for quantum channels [12, Lemma 8] that was first derived in [22]. Namely, we have for \(\omega _{C^n}\in S\left( {\mathcal {H}}_C^{\otimes n}\right) \) and permutation invariant \({\mathcal {R}}_{C^n\rightarrow B^nC^n}\) that

$$\begin{aligned} {\mathcal {R}}_{C^n\rightarrow B^nC^n}\left( \omega _{C^n}\right) \ll {{\,\mathrm{poly}\,}}(n)\int \left( {\mathcal {R}}_{C\rightarrow BC}\right) ^{\otimes n}\left( \omega _{C^n}\right) \mathrm {d}\nu ({\mathcal {R}}) \end{aligned}$$
(89)

for some measure \(\mathrm {d}\nu ({\mathcal {R}})\) over the completely positive and trace preserving maps on \(C\rightarrow BC\). As explained in the proof of [12, Proposition 9], the joint convexity of the quantum relative entropy together with the operator monotonicity of the logarithm then imply that

$$\begin{aligned}&D\left( \rho _{ABC}^{\otimes n}\big \Vert {\mathcal {R}}_{C^n\rightarrow B^nC^n}\left( \rho _{AC}^{\otimes n}\right) \right) \qquad \nonumber \\&\quad \ge D\Big (\rho _{ABC}^{\otimes n}\Big \Vert \int \big (({\mathcal {I}}_A\otimes {\mathcal {R}}_{C\rightarrow BC})(\rho _{AC})\big )^{\otimes n}\;\mathrm {d}\nu ({\mathcal {R}})\Big )-\log {{\,\mathrm{poly}\,}}(n)\,. \end{aligned}$$
(90)

\(\square \)

As such, we can conclude that

$$\begin{aligned} \zeta _{{\bar{{\mathcal {R}}}}}(\infty ,0)&:=\lim _{\varepsilon \rightarrow 0}\liminf _{n\rightarrow \infty }\zeta _{{\bar{{\mathcal {R}}}}}(n,\varepsilon ) \nonumber \\&= \lim _{\varepsilon \rightarrow 0}\limsup _{n\rightarrow \infty }\zeta _{{\bar{{\mathcal {R}}}}}(n,\varepsilon ) = D^{\infty }(A;B|C)_\rho \,. \end{aligned}$$
(91)

3.3 Quantum mutual information

The quantum mutual information of \(\rho _{AB}\in S({\mathcal {H}}_{AB})\) is defined as

$$\begin{aligned} I(A:B)_\rho :=H(A)_\rho +H(B)_\rho -H(AB)_\rho \,. \end{aligned}$$
(92)

Our main result from Sect. 2 provides a solution to the following discrimination problem.

Null hypothesis::

the fixed state \(\rho _{AB}^{\otimes n}\)

Alternative hypothesis::

the convex hull of iid states

$$\begin{aligned} {\bar{{\mathcal {T}}}}_{A^n:B^n}:=\Big \{\rho _A^{\otimes n}\otimes \int \sigma _B^{\otimes n}\;\mathrm {d}\mu (\sigma )\Big |\mu \in S({\mathcal {H}}_B)\Big \}\,. \end{aligned}$$
(93)

Namely, we have

$$\begin{aligned} {\bar{\zeta }}_{A:B}(\infty ,0):=&\lim _{\varepsilon \rightarrow 0}\lim _{n\rightarrow \infty }{\bar{\zeta }}_{A:B}(n,\varepsilon ) \end{aligned}$$
(94)
$$\begin{aligned} =&\lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\mu \in {\bar{{\mathcal {T}}}}}D\Big (\rho ^{\otimes n}_{AB}\Big \Vert \rho _A^{\otimes n}\otimes \int \sigma _B^{\otimes n}\;\mathrm {d}\mu (\sigma )\Big ) \end{aligned}$$
(95)
$$\begin{aligned} =&I(A:B)_\rho \,. \end{aligned}$$
(96)

Here, the last equality follows from the easily checked identity

$$\begin{aligned} I(A:B)_\rho =\inf _{\sigma _B\in S({\mathcal {H}})}D(\rho _{AB}\Vert \rho _A\otimes \sigma _B)\,. \end{aligned}$$
(97)

More general composite discrimination problems leading to the quantum mutual information were solved in [28] and in the following we further extend these results (cf. the classical work [53]).

Null hypothesis::

the fixed state \(\rho _{AB}^{\otimes n}\)

Alternative hypothesis::

the set of states

$$\begin{aligned} {\mathcal {T}}_{A^n:B^n}:=\left\{ \sigma _{A^n}\otimes \sigma _{B^n}\in S\left( {\mathcal {H}}_{AB}^{\otimes n}\right) \big |\sigma _{A^n}\;\text {or}\;\sigma _{B^n}\;\text {permutation invariant}\right\} \,. \end{aligned}$$
(98)

The goal is again the quantification of the optimal asymptotic error exponent

$$\begin{aligned} \zeta _{A:B}(n,\varepsilon )&:=-\frac{1}{n}\log \inf _{\begin{array}{c} 0\ll M_n\ll 1\\ {{\,\mathrm{Tr}\,}}\left[ M_n\rho ^{\otimes n}\right] \ge 1-\varepsilon \end{array}}\sup _{\sigma _{A^n}\otimes \sigma _{B^n}\in {\mathcal {T}}_n}{{\,\mathrm{Tr}\,}}\left[ M_{A^nB^n}\sigma _{A^n}\otimes \sigma _{B^n}\right] \end{aligned}$$
(99)
$$\begin{aligned} \mathrm {with}\quad \zeta _{A:B}(\infty ,\varepsilon )&:=\lim _{n\rightarrow \infty }\zeta _{A:B}(n,\varepsilon )\,. \end{aligned}$$
(100)

Note that the sets \({\mathcal {T}}_{A^nB^n}\) are not convex and hence the minimax technique used in Sect. 3.1 does not work here. However, following the ideas in [28, 53] we can exploit the permutation invariance and use de Finetti reductions of the form [15, 26] to find the following.

Proposition 3.4

For the discrimination problem as above with \(\varepsilon \in (0,1)\), we have

$$\begin{aligned} \zeta _{A:B}(\infty ,\varepsilon )=I(A:B)_\rho \,. \end{aligned}$$
(101)

The achievability direction is based on the following lemma.

Lemma 3.5

For the discrimination problem as above with \(n\in {\mathbb {N}}\) and \(\varepsilon \in (0,1)\), we have for \(s\in (0,1)\) that

$$\begin{aligned} \zeta _{A:B}(n,\varepsilon )\ge \inf _{\sigma \in S({\mathcal {H}})}D_s\left( \rho _{AB}\big \Vert \sigma _A\otimes \sigma _B\right) -\frac{1}{n}\frac{s}{1-s}\log \frac{1}{\varepsilon }-\frac{\log {{\,\mathrm{poly}\,}}(n)}{n}\,. \end{aligned}$$
(102)

Proof

Without loss of generality assume that \(\sigma _{A^n}\) is permutation invariant. We choose

$$\begin{aligned} M_{A^nB^n}(\lambda _n):=\left\{ \rho _{AB}^{\otimes n}-2^{\lambda _n}\omega _{A^n}\otimes \omega _{B^n}\right\} _+\quad \nonumber \\ \mathrm {with}\quad \omega _{A^n}:={n+|A|^2-1\atopwithdelims ()n}^{-1}{{\,\mathrm{Tr}\,}}_{{\tilde{A}}^n}\left[ P^{\mathrm {Sym}}_{A^n{\tilde{A}}^n}\right] \,, \end{aligned}$$
(103)

where \(P^{\mathrm {Sym}}_{A^n{\tilde{A}}^n}\) denotes the projector onto the symmetric subspace of \({\mathcal {H}}_A^{\otimes n}\otimes {\mathcal {H}}_{{\tilde{A}}}^{\otimes n}\) with \(|A|=|{\tilde{A}}|\) (denoting the dimension of \({\mathcal {H}}_A\) by |A|), and similarly for \(B^n\). Audenaert’s inequality (Lemma A.6) gives that

$$\begin{aligned}&{{\,\mathrm{Tr}\,}}\left[ (1-M_{A^nB^n}(\lambda _n))\rho ^{\otimes n}_{AB}\right] \le 2^{(1-s)\lambda _n}{{\,\mathrm{Tr}\,}}\left[ \left( \rho ^{\otimes n}_{AB}\right) ^s\left( \omega _{A^n}\otimes \omega _{B^n}\right) ^{1-s}\right] \nonumber \\&\quad \le 2^{(1-s)\left( \lambda _n-\inf _{\sigma _{A^n}\otimes \sigma _{B^n}\in {\mathcal {T}}_n}D_s\left( \rho ^{\otimes n}_{AB}\big \Vert \sigma _{A^n}\otimes \sigma _{B^n}\right) \right) }\,. \end{aligned}$$
(104)

Furthermore, we have by Schur-Weyl duality that \(\sigma _{A^n}\le {n+|A|^2-1\atopwithdelims ()n}\,\omega _{A^n}\) for all permutation invariant \(\sigma _{A^n}\) (see, e.g., [28, Lemma 1]) and thus again by Audenaert’s inequality (Lemma A.6)

$$\begin{aligned}&{{\,\mathrm{Tr}\,}}\left[ M_{A^nB^n}(\lambda _n)\left( \sigma _{A^n}\otimes \sigma _{B^n}\right) \right] \end{aligned}$$
(105)
$$\begin{aligned}&={{\,\mathrm{Tr}\,}}\left[ M_{A^nB^n}(\lambda _n)\left( \sigma _{A^n}\otimes \left( \sum _{\pi \in S_n}U_{B^n}(\pi )\sigma _{B^n}U_{B^n}^\dagger (\pi )\right) \right) \right] \;(S_n: \text { symm. group)}\nonumber \\&\le \underbrace{{n+|A|^2-1\atopwithdelims ()n}{n+|B|^2-1\atopwithdelims ()n}}_{=:\;p(n)\;\le \;{{\,\mathrm{poly}\,}}(n)}{{\,\mathrm{Tr}\,}}\left[ M_{A^nB^n}(\lambda _n)\left( \omega _{A^n}\otimes \omega _{B^n}\right) \right] \nonumber \\&\le p(n)\cdot 2^{-s\lambda _n}{{\,\mathrm{Tr}\,}}\left[ \left( \rho ^{\otimes n}_{AB}\right) ^s\left( \omega _{A^n}\otimes \omega _{B^n}\right) ^{1-s}\right] \nonumber \\&\le p(n)\cdot 2^{-s\lambda _n-(1-s)\inf _{\sigma _{A^n}\otimes \sigma _{B^n}\in {\mathcal {T}}_n}D_s\left( \rho ^{\otimes n}_{AB}\big \Vert \sigma _{A^n}\otimes \sigma _{B^n}\right) }\,. \end{aligned}$$
(106)

We now choose

$$\begin{aligned} \lambda _n:=\inf _{\sigma _{A^n}\otimes \sigma _{B^n}\in {\mathcal {T}}_n}D_s\left( \rho ^{\otimes n}_{AB}\big \Vert \sigma _{A^n}\otimes \sigma _{B^n}\right) +\log \varepsilon ^{\frac{1}{1-s}} \text { with }M_{A^nB^n}:=M_{A^nB^n}(\lambda _n), \end{aligned}$$
(107)

from which we get \({{\,\mathrm{Tr}\,}}\left[ M_{A^nB^n}\rho _{AB}^{\otimes n}\right] \ge 1-\varepsilon \) and together with Eqs. (99) and (106) that

$$\begin{aligned} \zeta _{A:B}^n(\varepsilon )\ge \inf _{\sigma _{A^n}\otimes \sigma _{B^n}\in {\mathcal {T}}_n}D_s\left( \rho _{AB}^{\otimes n}\big \Vert \sigma _{A^n}\otimes \sigma _{B^n}\right) -\frac{1}{n}\frac{s}{1-s}\log \frac{1}{\varepsilon }-\frac{\log p(n)}{n}\,. \end{aligned}$$
(108)

To deduce the claim it is now sufficient to argue that the Rényi quantum mutual informationFootnote 6

$$\begin{aligned} I_s(A:B)_\rho :=\inf _{\sigma _A\otimes \sigma _B\in S({\mathcal {H}})}D_s\left( \rho _{AB}\big \Vert \sigma _A\otimes \sigma _B\right) \end{aligned}$$
(109)

is additive on tensor product states. This, however, follows exactly as in the classical case [53, App. A-C] from the (quantum) Sibson identity [48, Lemma 3]

$$\begin{aligned}&D_s\left( \rho _{AB}\big \Vert \sigma _A\otimes \sigma _B\right) =D_s\left( \rho _{AB}\big \Vert \sigma _A\otimes {\bar{\sigma }}_B\right) +D_s\left( {\bar{\sigma }}_B\big \Vert \sigma _B\right) \quad \nonumber \\&\qquad \quad \mathrm {with}\quad {\bar{\sigma }}_B:=\frac{\left( {{\,\mathrm{Tr}\,}}_A\left[ \rho _{AB}^s\sigma _A^{1-s}\right] \right) ^\frac{1}{s}}{{{\,\mathrm{Tr}\,}}\left[ \left( {{\,\mathrm{Tr}\,}}_A\left[ \rho _{AB}^s\sigma _A^{1-s}\right] \right) ^\frac{1}{s}\right] }\,. \end{aligned}$$
(110)

\(\square \)

Taking the limit \(n\rightarrow \infty \) in Lemma 3.5 and then taking the limit \(s\rightarrow 1\) via the quantum Sibson identity from Eqs. (110) and  (97) yields

$$\begin{aligned} \lim _{s\rightarrow 1}I_s(A:B)_\rho =I(A:B)_\rho \,, \end{aligned}$$
(111)

gives the claimed achievability \(\zeta _{A:B}(\infty ,0)\ge I(A:B)_\rho \). A weak converse for \(\varepsilon \rightarrow 0\) follows as in Lemma 2.1 and the strong converse as claimed in Proposition 3.4 is derived similarly as in Proposition 3.1—by noting that it is sufficient to prove a converse for testing

$$\begin{aligned} \rho _{AB}^{\otimes n} \text { against }\rho _A^{\otimes n}\otimes \sigma _{B^n}. \end{aligned}$$
(112)

A more refined analysis of the above calculation along the work [28] allows to determine the Hoeffding bound for the product testing discrimination problem as above. However, for the strong converse exponent we are missing the additivity of the sandwiched Rényi quantum mutual information

$$\begin{aligned} {\tilde{I}}_s(A:B)_\rho :=\inf _{\sigma _ A\otimes \sigma _B\in S({\mathcal {H}})}{\tilde{D}}_s\left( \rho _{AB}\big \Vert \sigma _A\otimes \sigma _B\right) \end{aligned}$$
(113)

on product states.

4 Conditional Quantum Mutual Information

Here, we discuss how our results are related to the conditional quantum mutual information. This allows us to show that the regularization in our formula for composite asymmetric hypothesis testing as stated in Theorem 1.1 is needed in general.

4.1 Recoverability bounds

The following is a proof of the lower bound on the conditional quantum mutual information from Eq. (20).

Theorem 4.1

For \(\rho _{ABC}\in S({\mathcal {H}}_{ABC})\) we have

$$\begin{aligned} I(A:B|C)_\rho \ge \limsup _{n\rightarrow \infty }\frac{1}{n}D\Big (\rho _{ABC}^{\otimes n}\Big \Vert \int \beta _0(t)\left( {\mathcal {I}}_A\otimes {\mathcal {R}}^{[t]}_{C\rightarrow BC}(\rho _{AC})\Big )^{\otimes n}\mathrm {d}t\right) \,, \end{aligned}$$
(114)

where \({\mathcal {R}}^{[t]}_{C\rightarrow BC}(\cdot ):=\rho _{BC}^{\frac{1+it}{2}}\big (\rho _C^{\frac{-1-it}{2}}(\cdot )\rho _C^{\frac{-1+it}{2}}\big )\rho _{BC}^{\frac{1-it}{2}}\) with the inverses understood as generalized inverses and \(\beta _0(t):=\frac{\pi }{2}\left( \cosh (\pi t)+1\right) ^{-1}\).

Proof

We start from the lower bound [50, Theorem 4.1] applied to \(\rho _{ABC}^{\otimes n}\) (with the support conditions taken care of as in the corresponding proof)

$$\begin{aligned} I(A:B|C)_\rho =\frac{1}{n}I\left( A^n:B^n\big |C^n\right) _{\rho ^{\otimes n}}\ge \frac{1}{n}D_{{\mathcal {M}}}\left( \rho _{ABC}^{\otimes n}\big \Vert \sigma _{A^nB^nC^n}\right) \end{aligned}$$
(115)

with

$$\begin{aligned} \sigma _{A^nB^nC^n}:=\int \beta _0(t)\left( \sigma _{ABC}^{[t]}\right) ^{\otimes n}\mathrm {d}t \text { and }\sigma _{ABC}^{[t]}:=\left( {\mathcal {I}}_A\otimes {\mathcal {R}}^{[t]}_{C\rightarrow BC}\right) (\rho _{AC}), \end{aligned}$$
(116)

where we have used that the conditional quantum mutual information is additive on product states. Now, we simply observe that \(\sigma _{A^nB^nC^n}\) is permutation invariant and hence the claim can be deduced from Lemma 2.4 together with taking the limit superior \(n\rightarrow \infty \). \(\square \)

Together with previous work we find the following corollary that encompasses all known recoverability lower bounds on the conditional quantum mutual information.

Corollary 4.2

For \(\rho _{ABC}\in S({\mathcal {H}}_{ABC})\) the conditional quantum mutual information \(I(A:B|C)_\rho \) is lower bounded by

$$\begin{aligned}&-\int \beta _0(t)\log \Big \Vert \sqrt{\rho _{ABC}}\sqrt{\sigma _{ABC}^{[t]}}\Big \Vert _1^2\;\mathrm {d}t \end{aligned}$$
(117)
$$\begin{aligned}&D_{{\mathcal {M}}}\Big (\rho _{ABC}\Big \Vert \int \beta _0(t)\sigma _{ABC}^{[t]}\;\mathrm {d}t\Big ) \end{aligned}$$
(118)
$$\begin{aligned}&\limsup _{n\rightarrow \infty }\frac{1}{n}D\Big (\rho _{ABC}^{\otimes n}\Big \Vert \int \beta _0(t)\big (\sigma _{ABC}^{[t]}\big )^{\otimes n}\mathrm {d}t\Big ) \end{aligned}$$
(119)

with \(\sigma _{ABC}^{[t]}\) from Eq. (116).

The first bound was shown in [32, Section 3], the second one in [50, Theorem 4.1], and the third one is Theorem 4.1. We note that the lower bounds are typically strict in the non-commutative case, as can be seen from numerical work (see, e.g., [12]). In contrast to the second and third bound, the first lower bound is not tight in the commutative case but has the advantage that the average over \(\beta _0(t)\) stands outside of the distance measure used. Moreover, the distribution \(\beta _0(t)\) cannot be taken outside the relative entropy measure in the second and the third bound, since quantum Stein’s lemma would then lead to a contradiction to a recent counterexample from [21, Section 5]. Namely, there exists \(\theta \in \left[ 0,\pi /2\right] \) such that

$$\begin{aligned}&I(A:B|C)_\rho \ngeq \inf _{{\mathcal {R}}}D\left( \rho _{ABC}\big \Vert ({\mathcal {I}}_A\otimes {\mathcal {R}}_{C\rightarrow BC})(\rho _{AC})\right) \; \end{aligned}$$
(120)

for the pure state \(\rho _{ABC}=|\rho \rangle \langle \rho |_{ABC}\) with

$$\begin{aligned} |\rho \rangle _{ABC}=&\frac{1}{\sqrt{2}}\big (\cos (\theta )|0\rangle _A\otimes |1\rangle _C+\sin (\theta )|1\rangle _A\otimes |0\rangle _C\big )\otimes |1\rangle _B\nonumber \\&+\frac{1}{\sqrt{2}}|0\rangle _A\otimes |0\rangle _B\otimes |0\rangle _C\,. \end{aligned}$$
(121)

It seems that the only remaining conjectured strengthening is the lower bound in terms of the non-rotated Petz map [8, Section 8]

$$\begin{aligned} I(A:B|C)_\rho \ge -\log \Big \Vert \sqrt{\rho _{ABC}}\sqrt{\sigma _{ABC}^{[0]}}\Big \Vert _1^2\,. \end{aligned}$$
(122)

We refer to [33] for the latest progress in that direction.

The arguments in this section can also be applied to lift the strengthened monotonicity from [50, Corollary 4.2]. For \(\rho \in S({\mathcal {H}})\), \(\sigma \) a positive semi-definite operator on \({\mathcal {H}}\), and \({\mathcal {N}}\) a completely positive trace preserving map on the same space this leads to

$$\begin{aligned} D(\rho \Vert \sigma )-D({\mathcal {N}}(\rho )\Vert {\mathcal {N}}(\sigma ))\ge \limsup _{n\rightarrow \infty }\frac{1}{n}D\Big (\rho ^{\otimes n}\Big \Vert \int \beta _0(t)\left( {\mathcal {R}}^{[t]}_{\sigma ,{\mathcal {N}}}(\rho )\right) ^{\otimes n}\mathrm {d}t\Big ), \end{aligned}$$
(123)

where \({\mathcal {R}}^{[t]}_{\sigma ,{\mathcal {N}}}(\cdot ):=\sigma ^{\frac{1+it}{2}}{\mathcal {N}}^{\dagger }\left( {\mathcal {N}}(\sigma )^{\frac{-1-it}{2}}(\cdot ){\mathcal {N}}(\sigma )^{\frac{-1+it}{2}}\right) \sigma ^{\frac{1-it}{2}}\). Together with [32, Section 3] and [50, Corollary 4.2] we then again have the three lower bounds as in Corollary 4.2.

4.2 Regularization necessary

Here, we use our bound on the conditional quantum mutual information (Theorem 4.1) to show that the regularization in Theorem 1.1 is in general needed (see also [10]). That is, we give a proof for Eq. (17). Namely, by Theorem 4.1 we haveFootnote 7

$$\begin{aligned} I(A:B|C)_\rho&\ge \limsup _{n\rightarrow \infty }\frac{1}{n}D\Big (\rho _{ABC}^{\otimes n}\Big \Vert \int \beta _0(t)\left( {\mathcal {I}}_A\otimes {\mathcal {R}}^{[t]}_{C\rightarrow BC}(\rho _{AC})\right) ^{\otimes n}\mathrm {d}t\Big ) \end{aligned}$$
(124)
$$\begin{aligned}&\ge \lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\mu \in {\mathcal {R}}}D\Big (\rho _{ABC}^{\otimes n}\Big \Vert \int \left( {\mathcal {I}}_A\otimes {\mathcal {R}}_{C\rightarrow BC}\left( \rho _{AC}\right) \right) ^{\otimes n}\mathrm {d}\mu ({\mathcal {R}})\Big )\,. \end{aligned}$$
(125)

From the second composite discrimination problem described in Sect. 3.2 we see that the latter quantity is equal to the asymptotic error exponent \(\zeta _{{\bar{{\mathcal {R}}}}}(\infty ,0)\) as given in Eq. (91) for testing

$$\begin{aligned} \rho _{ABC}^{\otimes n} \text { against } \int \left( ({\mathcal {I}}_A\otimes {\mathcal {R}}_{C\rightarrow BC})(\rho _{AC})\right) ^{\otimes n}\;\mathrm {d}\mu ({\mathcal {R}}). \end{aligned}$$
(126)

Now, if the regularization in the asymptotic formula for \(\zeta _{{\bar{{\mathcal {R}}}}}(\infty ,0)\) would actually not be needed this would imply that

$$\begin{aligned} I(A:B|C)_\rho \ge \inf _{{\mathcal {R}}}D\left( \rho _{ABC}\Vert ({\mathcal {I}}_A\otimes {\mathcal {R}}_{C\rightarrow BC})(\rho _{AC})\right) \,. \end{aligned}$$
(127)

However, this is in contradiction with the counterexample from [21, Section 5] as discussed in Eq. (120). Hence, we conclude that the regularization for composite asymmetric quantum hypothesis testing is needed in general. \(\square \)

5 Conclusion

We extended quantum Stein’s lemma in asymmetric quantum hypothesis testing by showing that the optimal asymptotic error exponent for testing convex combinations of quantum states \(\rho ^{\otimes n}\) against convex combinations of quantum states \(\sigma ^{\otimes n}\) is given by a regularized quantum relative entropy formula which does not become single-letter in general. Moreover, we gave various examples when our formula as well as extensions thereof become single-letter. It remains interesting to find more non-commutative settings that allow for single-letter solutions.

Another related problem is that of symmetric hypothesis testing, where it is well-known that in the case of fixed iid states \(\rho ^{\otimes n}\) against \(\sigma ^{\otimes n}\) the optimal asymptotic error exponent is given by the quantum Chernoff bound [1, 42]

$$\begin{aligned} C(\rho ,\sigma ):=\sup _{0\le s\le 1}-\log {{\,\mathrm{Tr}\,}}\left[ \rho ^s\sigma ^{1-s}\right] \,. \end{aligned}$$
(128)

For this symmetric setting, it was conjectured in [2] that for finite sets \({\mathcal {S}}\) and \({\mathcal {T}}\) the corresponding composite asymptotic error exponent is given by

$$\begin{aligned} C({\mathcal {S}},{\mathcal {T}}):=\inf _{\begin{array}{c} \rho \in {\mathcal {S}}\\ \sigma \in {\mathcal {T}} \end{array}}C(\rho ,\sigma )\,, \end{aligned}$$
(129)

with definitions analogue to those given earlier for the asymmetric setting. However, it was recently shown that already in the setting of a fixed null hypothesis \({\mathcal {S}}=\{\rho \}\) above conjecture is in general false [38].Footnote 8

Moreover, one can again consider testing convex combinations of iid states \(\rho ^{\otimes n}\) with \(\rho \in {\mathcal {S}}\) against convex combinations of iid states \(\sigma ^{\otimes n}\) with \(\sigma \in {\mathcal {T}}\). Similarly to our work for the asymmetric setting, we then have that the following rate for the asymptotic error exponent is achievable (assuming that the limit exists)

$$\begin{aligned} \sup _{0\le s\le 1} \lim _{n\rightarrow \infty }\frac{1}{n}\inf _{\begin{array}{c} \nu \in {\mathcal {S}}\\ \mu \in {\mathcal {T}} \end{array}}-\log {{\,\mathrm{Tr}\,}}\Big [\Big (\int \rho ^{\otimes n}\;\mathrm {d}\nu (\rho )\Big )^s\Big (\int \sigma ^{\otimes n}\;\mathrm {d}\mu (\sigma )\Big )^{1-s}\Big ]\,. \end{aligned}$$
(130)

However, it was already shown in [29] that this does in general not simplify to the single-letter form in Eq. (129). We refer to [38, Section I] for an excellent overview of the recent progress on composite hypothesis testing.

Finally, we note that finding single-letter achievability results for composite hypothesis testing problems has important applications in network quantum Shannon theory [45, Section 5.2].